Find the number of different 8-letter arrange | vedclass

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Question

(A) The word $DAUGHTER$ consists of $8$ distinct letters: $D, A, U, G, H, T, E, R$.There are $3$ vowels: $A, U, E$ and $5$ consonants: $D, G, H, T, R$

Vedclass · Accepted Answer

$4320$

Vedclass · Answer

(A) The word $DAUGHTER$ consists of $8$ distinct letters: $D, A, U, G, H, T, E, R$.There are $3$ vowels: $A, U, E$ and $5$ consonants: $D, G, H, T, R$.Since the vowels must occur together,we treat them as a single unit or object: $(AUE)$.Now,we have $5$ consonants plus $1$ unit of vowels,making a total of $6$ objects.These $6$ objects can be arranged in $6!$ ways.Within the vowel unit,the $3$ vowels $(A, U, E)$ can be arranged among themselves in $3!$ ways.By the multiplication principle,the total number of arrangements is $6! 	imes 3!$.$6! = 720$ and $3! = 6$.Total arrangements $= 720 	imes 6 = 4320$.

Find the number of different $8$-letter arrangements that can be made from the letters of the word $DAUGHTER$ such that all vowels occur together.

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